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Theorem cleljust 2153
Description: When the class variables in definition df-clel 2767 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2146 with the class variables in wcel 2145. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2090 in order to remove dependencies on ax-10 2174, ax-12 2203, ax-13 2408. Note that there is no DV condition on 𝑥, 𝑦, that is, on the variables of the left-hand side. (Revised by BJ, 29-Dec-2020.)
Assertion
Ref Expression
cleljust (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljust
StepHypRef Expression
1 elequ1 2152 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
21equsexvw 2090 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
32bicomi 214 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by:  bj-dfclel  33218
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